Signals and Systems - Electrical Engineering

5.4 Fourier Transforms from the Laplace Transform 303

The following rules of thumb will help you get a better understanding of the time-frequency relation-
ship of a signal and its Fourier transform, and the best way to compute it. On a first reading the use
of these rules might not be obvious, but they will be helpful in understanding the discussions that
follow and you might want to come back to these rules.

Rules of Thumb for Computing the Fourier Transform of a Signalx(t)

n Ifx(t)has a finite time support and in that supportx(t)is finite, its Fourier transform exists. To find it use the integral

definition or the Laplace transform ofx(t).

n Ifx(t)has a Laplace transformX(s)with a region of convergence including thejaxis, its Fourier transform isX(s)|s=j.

n Ifx(t)is periodic of infinite energy but finite power, its Fourier transform is obtained from its Fourier series using delta

functions.

n Ifx(t)is none of the above, if it has discontinuities (e.g.,x(t)=u(t)) or it has discontinuities and it is not finite energy

(e.g.,x(t)=cos( 0 t)u(t)), or it has possible discontinuities in the frequency domain even though it has finite energy

(e.g.,x(t)=sinc(t)), use properties of the Fourier transform.

Keep in mind to

n Consider the Laplace transform if the interest is in transients and steady state, and the Fourier
transform if steady-state behavior is of interest.
n Represent periodic signals by their Fourier series before considering their Fourier transforms.
n Attempt other methods before performing integration to find the Fourier transform.

nExample 5.1

Discuss whether it is possible to obtain the Fourier transform of the following signals using their

Laplace transforms:

(a) x 1 (t)=u(t)

(b)x 2 (t)=e−^2 tu(t)

(c) x 3 (t)=e−|t|

Solution

(a) The Laplace transform ofx 1 (t)isX 1 (s)= 1 /swith a region of convergence corresponding

to the open rights-plane, or ROC={s=σ+j:σ >0,−∞<  <∞}, which does not

include thejaxis, so the Laplace transform cannot be used to find the Fourier transform

ofx 1 (t).

(b) The signalx 2 (t)has as Laplace transformX 2 (s)= 1 /(s+ 2 )with a region of convergence ROC

={s=σ+j:σ >−2,−∞<  <∞}containing thejaxis. Then the Fourier transform of

x 2 (t)is

X 2 ()=

1

s+ 2

∣

∣s=j=^1

j+ 2